But how he achieves that remains to be seen. The Senate Finance Committee stipulated that federal funds could not be used for abortions but that insurance plans for people who receive federal subsidies could cover abortion as long as they were paid for with premiums from patients and not federal subsidies.
Conservative Democrats in the House, with the help of Republicans, were able to dismiss such a "firewall" approach for an all-out ban on abortion coverage for any plan that receives subsidies.
"I laid out a very simple principle, which is this is a health care bill, not an abortion bill," Obama said. "And we're not looking to change what is the principle that has been in place for a very long time, which is federal dollars are not used to subsidize abortions."
Obama said he was confident that the final legislation will ensure that "neither side feels that it's being betrayed.
"I want to make sure that the provision that emerges meets that test; that we are not in some way sneaking in funding for abortions, but, on the other hand, that we're not restricting women's insurance choices," he said.
Abortion is only one of the internal disputes that have snarled Democrats over health reform. Another issue that could threaten Democrats' ability to pass a health care bill through the Senate is public option, a government-run insurance plan that would compete with the private sector.
Reid said he spoke today with Sen. Joe Lieberman, the independent Democrat from Connecticut who has threatened to filibuster a health care bill if it includes a public option, and Reid said he is confident "we can get something done."
Medicare cuts are another thorny issue. Asked whether the Medicare cuts he proposes to help pay for health care overhaul would be undone by Congress subsequently, as is often the case with deficit and cost-cutting measures, the president told ABC News he will pledge to undo any such measures.
"I actually have said that it is important for us to make sure this thing is deficit neutral, without tricks," he said. "What I also said in that speech to the joint session was that I'm willing to put in some safeguards where if we don't obtain the savings that have been promised, that we've got to make adjustments in terms of the benefits, because the goal here is to reduce costs for families, give them more security, but do so in a way that is not adding to our deficit ... Congress needs to know that when I say this has to be deficit neutral, I mean it."
Despite the difficulty of reaching accord on abortion and the public option, when Reid was asked by reporters if he could get a bill on the Senate floor by next week and pass it out of the Senate by Christmas, he said, "Yes and yes."
Reid was speaking at a new conference in the Capitol Building with veterans groups and fellow Democrats to bring pressure on Sen. Tom Coburn, R-Okla, who has blocked a bill to authorize new programs for the care of veterans who need in-home care.
Coburn has said he wants new spending in the bill to be accounted for by cuts elsewhere in the budget.
But Democrats said today that funding for veterans was too important to slow down.